Standard and Non-standard Extensions of Lie algebras

TL;DR

This paper explores quadruple extensions of simple Lie algebras, revealing limitations of standard extensions and proposing non-standard methods, including folding procedures, to construct new algebraic structures relevant to theoretical physics.

Contribution

It introduces non-standard extension procedures for Lie algebras and explicitly constructs certain high-rank subalgebras of E11, expanding the understanding of algebraic extensions.

Findings

Quadruple extensions with simple links are not possible unless using Borcherds imaginary roots.

Folding procedures generate all non-simply laced triple extended Lie algebras.

Explicit root systems of rank 11 subalgebras of E11 containing E10 are provided.

Abstract

We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root $\alpha_{+4}$, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if $\alpha_{+4}$ is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply-laced triple extended Lie algebras, obtaining all the non-simply laced ones. Non- standard extension procedures for a class of Lie algebras are proposed. It is shown that the 2-extensions of $E_{8}$, with a dot simply linked to the Dynkin-Kac diagram of $E_{9}$, are rank 10 subalgebras of $E_{10}$. Finally the simple root systems of a set of rank 11 subalgebras of $E_{11}$, containing as sub-algebra $E_{10}$, are explicitly written.